✨Danh sách tích phân với hàm lượng giác ngược

Dưới đây là danh sách các tích phân với hàm lượng giác ngược.

: $\backslash int\backslash arcsin\backslash frac\{x\}\{c\}\backslash ,dx\; =\; x\backslash arcsin\backslash frac\{x\}\{c\}\; +\; \backslash sqrt\{c^2-x^2\}$

: $\backslash int\; x\; \backslash arcsin\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash left(\backslash frac\{x^2\}\{2\}-\backslash frac\{c^2\}\{4\}\backslash right)\backslash arcsin\backslash frac\{x\}\{c\}\; +\; \backslash frac\{x\}\{4\}\backslash sqrt\{c^2-x^2\}$

: $\backslash int\; x^2\; \backslash arcsin\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{3\}\backslash arcsin\backslash frac\{x\}\{c\}\; +\; \backslash frac\{x^2+2c^2\}\{9\}\backslash sqrt\{c^2-x^2\}$

: $\backslash int\; x^n\; \backslash sin^\{-1\}x\backslash ,dx\; =\; \backslash frac\{1\}\{n+1\}\backslash left(x^\{n+1\}\backslash sin^\{-1\}x\; \backslash right.$ :::$\backslash left.\; +\; \backslash frac\{x^n\backslash sqrt\{1\; -\; x^2\}\; -\; nx^\{n-1\}\backslash sin^\{-1\}x\}\{n-1\}\; +\; n\backslash int\; x^\{n-2\}\backslash sin^\{-1\}x\backslash ,dx\backslash right)$

: $\backslash int\backslash arccos\backslash frac\{x\}\{c\}\backslash ,dx\; =\; x\backslash arccos\backslash frac\{x\}\{c\}\; -\; \backslash sqrt\{c^2-x^2\}$

: $\backslash int\; x\; \backslash arccos\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash left(\backslash frac\{x^2\}\{2\}-\backslash frac\{c^2\}\{4\}\backslash right)\backslash arccos\backslash frac\{x\}\{c\}\; -\; \backslash frac\{x\}\{4\}\backslash sqrt\{c^2-x^2\}$

: $\backslash int\; x^2\; \backslash arccos\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{3\}\backslash arccos\backslash frac\{x\}\{c\}\; -\; \backslash frac\{x^2+2c^2\}\{9\}\backslash sqrt\{c^2-x^2\}$

: $\backslash int\backslash arctan\backslash frac\{x\}\{c\}\backslash ,dx\; =\; x\backslash arctan\backslash frac\{x\}\{c\}\; -\; \backslash frac\{c\}\{2\}\backslash ln(c^2+x^2)$

: $\backslash int\; x\; \backslash arctan\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{c^2+x^2\}\{2\}\backslash arctan\backslash frac\{x\}\{c\}\; -\; \backslash frac\{cx\}\{2\}$

: $\backslash int\; x^2\; \backslash arctan\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{3\}\backslash arctan\backslash frac\{x\}\{c\}\; -\; \backslash frac\{cx^2\}\{6\}\; +\; \backslash frac\{c^3\}\{6\}\backslash ln\{c^2+x^2\}$

: $\backslash int\; x^n\; \backslash arctan\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^\{n+1\{n+1\}\backslash arctan\backslash frac\{x\}\{c\}\; -\; \backslash frac\{c\}\{n+1\}\backslash int\backslash frac\{x^\{n+1\}\; dx\}\{c^2+x^2\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\; \backslash arcsec\{\backslash frac\{x\}\{c\backslash ,dx\; =\; x\; \backslash arcsec\{\backslash frac\{x\}\{c\; +\; \backslash frac\{x\}\{c|x|\}\backslash ln$

: $\backslash int\; x\backslash arcsec\{x\}\backslash ,dx\backslash ,=\backslash ,\backslash frac\{1\}\{2\}\backslash left(x^2\backslash arcsec\{x\}\; -\; \backslash sqrt\{x^2\; -\; 1\}\backslash right)$

: $\backslash int\; x^n\backslash arcsec\{x\}\backslash ,dx\backslash ,=\backslash ,\backslash frac\{1\}\{n+1\}\backslash left(x^\{n+1\}\backslash arcsec\{x\}\; -\; \backslash frac\{1\}\{n\}\backslash left(x^\{n-1\}\backslash sqrt\{x^2\; -\; 1\}\backslash ;\; \backslash right.\; \backslash right.$ ::: $\backslash left.\; \backslash left.\; +\; (1-n)\backslash left(x^\{n-1\}\backslash arcsec\{x\}\; +\; (1-n)\backslash int\; x^\{n-2\}\backslash arcsec\{x\}\backslash ,dx\; \backslash right)\backslash right)\backslash right)$

: $\backslash int\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\backslash ,dx\; =\; x\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\; +\; \backslash frac\{c\}\{2\}\backslash ln(c^2+x^2)$

: $\backslash int\; x\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{c^2+x^2\}\{2\}\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\; +\; \backslash frac\{cx\}\{2\}$

: $\backslash int\; x^2\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{3\}\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\; +\; \backslash frac\{cx^2\}\{6\}\; -\; \backslash frac\{c^3\}\{6\}\backslash ln(c^2+x^2)$

: $\backslash int\; x^n\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\backslash ,dx\; =\; \backslash frac\{x^\{n+1\{n+1\}\backslash ,\backslash mathrm\{arccot\}\backslash ,\backslash frac\{x\}\{c\}\; +\; \backslash frac\{c\}\{n+1\}\backslash int\backslash frac\{x^\{n+1\}\; dx\}\{c^2+x^2\}\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$